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Citation: BATCHELOR, S., KEEBLE, S. and GILMORE, C.K., 2015. Magnitude representations and counting skills in preschool children. Mathematical
Thinking and Learning, 17 (2-3), pp.116-135.
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Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
Magnitude Representations and Counting Skills in Preschool Children
Sophie Batchelor, Sarah Keeble and Camilla Gilmore
Mathematics Education Centre, Loughborough University
Author Note
This research was supported in part by a British Academy Fellowship awarded to
Camilla Gilmore.
Correspondence concerning this article should be addressed to Sophie Batchelor,
Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire,
LE11 3TU, UK. E-mail: [email protected]
1
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
2
Abstract
When children learn to count, they map newly acquired symbolic representations of number
onto preexisting nonsymbolic representations. The nature and timing of this mapping is
currently unclear. Some researchers have suggested this mapping process helps children
understand the cardinal principle of counting, while other evidence suggests that this
mapping only occurs once children have cardinality understanding. One difficulty with the
current literature is that studies have employed tasks that only indirectly assess children’s
nonsymbolic-symbolic mappings. We introduce a task in which preschoolers made
magnitude comparisons across representation formats (e.g., dot arrays vs. verbal number),
allowing a direct assessment of mapping. We gave this task to 60 children aged 2;7 - 4;10,
together with counting and Give-a-Number tasks. We found that some children could map
between nonsymbolic quantities and the number words they understood the cardinal meaning
of, even if they had yet to grasp the general cardinality principle of counting.
Keywords: counting, magnitude comparison, cardinality, preschool children, number
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
3
Magnitude Representations and Counting Skills in Preschool Children
We know from more than a decade’s worth of research that infants, children and adults
can represent and manipulate numerical information nonsymbolically, without number words
or digits. These nonsymbolic representations are robust across multiple modalities and set
sizes. Children and adults can compare, add and subtract small and large quantities in visual
arrays (Barth, Kanwisher, & Spelke, 2003; McCrink & Wynn, 2004), auditory sequences
(Barth et al., 2003; Barth, La Mont, Lipton, & Spelke, 2005) and moving displays of actions
(e.g., puppet jumps) (Wood & Spelke, 2005; Wynn, 1996; Wynn, Bloom, & Chiang, 2002).
The nonsymbolic representations employed in these tasks are approximate and in an
analogue format. They are inherently noisy and the variance associated with them increases
with the absolute size of the magnitudes represented. As a result, success on these tasks
depends on the ratio (or numerical distance) between the numerosities to be compared1. As
the quantities get closer together, discrimination becomes more effortful and less precise.
Importantly, the precision of these representations varies across individuals and increases
over development. Infants can discriminate numerosities with ratios as small as 2:3, whilst
preschool children show a ratio-limit of 3:4, and adults, 7:8 (Barth et al., 2003; Feigenson,
Dehaene, & Spelke, 2004).
When children begin to count they learn to use external symbols to represent number.
These symbolic representations enable exact number comparison and manipulation. There is
evidence that when children acquire this symbolic system, the preexisting nonsymbolic
system is not overridden; rather, nonsymbolic representations become mapped onto the
newly acquired symbolic representations. The evidence for this is at least threefold. Firstly,
children and adults show a numerical distance effect for symbolic number comparison
(Dehaene, Dupoux, & Mehler, 1990; Moyer & Landauer, 1967; Temple & Posner, 1998).
When asked to compare two symbolic numbers (e.g., Arabic digits) reaction times and
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
4
accuracy vary systematically with the numerical distance between the symbols. As with
nonsymbolic comparison, performance is slower and less accurate as the numbers to be
compared get closer. This suggests that symbolic numbers are mapped onto, and
automatically activate, approximate nonsymbolic representations. Secondly, evidence for
these mappings come from investigations showing that children and adults can perform rapid
approximate arithmetic on symbolic numbers (e.g., Gilmore, McCarthy, & Spelke, 2007;
Xenidou-Dervou, De Smedt, van der Schoot, & van Lieshout, 2013). These approximate
computations engage a distinct neural system to that which is activated during exact symbolic
arithmetic (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). Thirdly, some
neuropsychological studies have revealed associations between the nonsymbolic and
symbolic systems (Dehaene, Dehaene-Lambertz, & Cohen, 1998). Impairments in
nonsymbolic processing are associated with impairments in the symbolic domain, and vice
versa.
This evidence suggests that, in individuals who have already acquired the symbolic
system, symbolic representations are connected with preexisting nonsymbolic
representations. However, we do not know how these symbolic-nonsymbolic mappings are
formed, or whether they play a role in the acquisition of symbolic mathematical skills. Below
we review evidence concerning the relationship between nonsymbolic and symbolic
representations and mathematics, before turning to evidence concerning how these mappings
are formed.
Magnitude Representations and Mathematics
Children and adults show individual differences in the precision of their numerical
magnitude representations (Barth et al., 2003; Feigenson et al., 2004). The precision of these
representations is typically indexed by performance on magnitude comparison tasks, either
nonsymbolic (dot comparison) or symbolic (Arabic digit comparison). Participants are
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
5
presented with two numerosities (dot arrays or Arabic digits) and are asked to select the more
numerous. Several researchers have explored the relationship between children’s and adults’
performance on these tasks with performance on standardised tests of mathematics
achievement. The evidence for a relationship between mathematics and nonsymbolic
comparison performance has been somewhat mixed (see meta-analyses by Chen & Li, 2014;
Fazio, Bailey, Thompson, & Siegler, 2014), although there is more consistent evidence for a
relationship between mathematics and symbolic comparison performance (De Smedt, Noël,
Gilmore, & Ansari, 2013).
These findings have been taken to suggest that the mapping between nonsymbolic and
symbolic representations may play a more pivotal role in later mathematics achievement than
nonsymbolic acuity itself. It is not clear, however, whether the symbolic comparison task is
measuring the mapping between nonsymbolic and symbolic representations, or whether it is
measuring the nature of nonsymbolic or symbolic skills per se. To understand the nature of
the mapping between representations and importantly, how this mapping is related to formal
mathematics, we need more direct measures of mapping ability. Mundy and Gilmore (2009)
introduced a novel mapping task in which children had to decide which of two nonsymbolic
arrays matched a symbolic numerosity (symbolic to nonsymbolic mapping) or which of two
symbolic numerosities matched a nonsymbolic array (nonsymbolic to symbolic mapping).
Here it was found that individual differences in mapping ability accounted for variation in
school mathematics achievement over and above standard symbolic and nonsymbolic
numerical comparison tasks. Similarly Brankaer, Ghesquière, and De Smedt (2014) found
that children’s performance on a similar mapping task developed from age 6 to 8 and was
related to performance on both timed and untimed mathematics tests, after controlling for
magnitude comparison performance. These findings concur with data from Booth and Siegler
(2008), which showed a relationship between children’s ability to map symbolic
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
6
representations onto a number line and later arithmetical achievement. These findings raise
the question whether nonsymbolic representations and the mapping between nonsymbolic
and symbolic representations play an instrumental role in the acquisition of symbolic
representations. We turn to evidence on this point in the next section.
Magnitude Representations and Learning to Count
Several researchers have explored whether the formation of mappings between symbolic
and nonsymbolic representations are critical for the acquisition of symbolic number
knowledge (e.g., Le Corre & Carey, 2007; Wagner & Johnson, 2011). One possibility is that
when children learn to count, they map newly acquired symbolic representations onto their
preexisting nonsymbolic representations, and that the nonsymbolic system and in particular
this mapping process helps with the acquisition of numbers (e.g., Gallistel & Gelman, 1992).
Alternatively, the symbolic system may be acquired independently of the nonsymbolic
system and the mapping between nonsymbolic and symbolic representations might occur
later, once children have acquired the symbolic system (e.g., Le Corre & Carey, 2007). There
is currently evidence for both of these accounts.
Before we review this evidence, we briefly describe three tasks typically used to measure
children’s counting skills and knowledge. Count list elicitation (or ‘How high?’) tasks are
used to assess children’s ability to generate the words in the count list. In some versions of
this task children are simply asked to count up as high as they can and are scored based on
the highest number they recall before making any errors (e.g., Barth, Starr, & Sullivan, 2009;
Lipton & Spelke, 2005). In other versions of the task children are provided with a set of
objects and are asked to count them (e.g., Le Corre & Carey, 2007; Slaughter, Kamppi, &
Paynter, 2006). Meanwhile, ‘How many?’ and ‘Give-a-Number’ tasks are used to assess
whether children understand the meaning of the words in the count list. They are assumed to
measure children’s knowledge of cardinality; i.e. understanding that the last word in their
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
7
count list represents the numerosity of the set as a whole (Gelman & Gallistel, 1978). In the
‘How many?’ task children are presented with a set of objects and they are asked to tell the
researcher how many there are (unlike a count elicitation task with objects, children are not
given the instruction to count the objects). In the ‘Give-a-Number’ task children are asked to
give the researcher a requested number of objects. In both of these tasks children receive a
score based on the highest number they can reliably produce.
A number of studies have explored the relationship between children’s ability to compare
nonsymbolic representations and their knowledge of the symbolic number system. HuntleyFenner and Cannon (2000) gave children aged 3-5 years a nonsymbolic (dot) comparison task
and two to four weeks later they assessed verbal counting ability. Counting was assessed with
a ‘How high?’ task and a ‘How many?’ task involving sets of 1-15 cubes. In line with
previous studies, children’s performance on the dot comparison task varied as a function of
numerical ratio, thus demonstrating the signature ratio-effect of the nonsymbolic system.
Correlation analyses revealed that the size of this ratio-effect was related to children’s
performance on the ‘How high?’ task, but not the ‘How many?’ task. Children with smaller
ratio-effects, and therefore more precise nonsymbolic representations, were better able to
recite the number words in their count list; however, they were no better able to understand
the meanings of these words. Further to this, data from Slaughter et al., (2006) showed that
preschool children’s performance on a nonsymbolic subtraction task was unrelated to their
ability to enumerate sets of up to 30 stickers. Together, these findings were taken to suggest
that nonsymbolic magnitude representations develop independently of children’s number
word mappings and therefore the nonsymbolic system does not play a role in helping children
acquire symbolic number.
More recent research has questioned this conclusion. Wagner and Johnson (2011)
highlighted some methodological limitations of the counting tasks used in these studies.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
8
Specifically, they emphasised that children’s success on a ‘How many?’ task does not
necessarily reflect an understanding of the counting system. Children may learn to report the
final tag of a count in response to ‘How many?’ without any conceptual understanding of
cardinality. Indeed, previous research has shown that the ‘How many?’ task is a less reliable
measure of whether children understand the counting system than a ‘Give-a-Number’ task in
which children are asked to produce sets of items (Wynn, 1992). In view of this, Wagner and
Johnson provided a further test of the relationship between children’s nonsymbolic skills and
their cardinality understanding using a variation of the ‘Give-a-Number’ task. In contrast to
previous studies, results revealed a positive correlation between children’s performance on
the nonsymbolic comparison task and their cardinality understanding. Mussolin, Nys,
Leybaert, and Content (2012) also found an association between children’s nonsymbolic
acuity and performance on a ‘Give-a-Number’ task. Interestingly, this correlation was
significant for the younger children, aged 3-4 years, but not the older children, aged 5-6
years. This suggests that the role of nonsymbolic representations may change over
development and highlights the importance of testing these abilities in children who are at the
start of learning counting skills.
The studies by Wagner and Johnson (2011) and Mussolin et al. (2012) suggest that the
precision of the nonsymbolic system is related to children’s counting ability and therefore the
nonsymbolic system may be involved in the acquisition of the symbolic number system. To
measure nonsymbolic skills, these studies have used dot comparison tasks. Despite being
widely used, dot comparison tasks have yet to be standardised making it difficult to compare
the results across studies. One issue is that studies have employed different controls over the
nonnumerical aspects of the dot stimuli. Some studies control for dot size and envelope area
whilst others control for only one (or none) of these factors. Wagner and Johnson (2011), for
example, only controlled for area. They note that children may have solved the task by
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
9
simply adopting a strategy of selecting the array with the smaller dots. This makes it difficult
to draw conclusions from their study. More recent research has suggested that inhibitory
control processes can affect an individual’s dot comparison performance (Gilmore et al.,
2013). Therefore, any relationship between nonsymbolic comparison performance and
symbolic number skills may be an artefact of nonnumerical task features. Moreover, to allow
sufficient control for visual features of the displays, dot comparison tasks typically require a
large number of trials to provide a reliable estimate of nonsymbolic acuity (e.g., Inglis &
Gilmore, 2014, suggest a minimum of 80 trials). This makes the dot comparison task
unsuitable for use with young children.
Turning to studies that have employed tasks to measure symbolic-nonsymbolic
mappings, rather than measures of the nonsymbolic system itself, there remain mixed
findings. Lipton and Spelke (2005) studied preschool children’s mapping of large number
words to nonsymbolic numerosities. They gave children a counting assessment together with
three tasks designed to measure knowledge of number word mappings: 1) An estimation
(‘How many on this card?’) task, 2) a number word comprehension task and 3) a number
word ordering task. Results showed that children could only map number words for those
numbers within their counting proficiency level. Children who failed to count beyond 60
failed estimation, comprehension and ordering tasks for number words larger than 60. The
authors conclude that children map nonsymbolic and symbolic number representations at
around the time that they master the count sequence. However, the children in this study were
aged between 5 years and 6 years 2 months and are likely to have had some years of
experience with counting.
In a follow-up study, Barth et al., (2009) examined the mappings of children with less
counting knowledge. Here they found that whilst the least skilled counters did not produce
accurate estimates on an estimation task, they did produce significantly larger estimates for
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
10
larger sets. This suggests that even very inexperienced counters have some knowledge of
large number word mappings. Similarly, Wagner and Johnson (2011) found that preschool
children showed scalar variability in their mappings of number words that were beyond their
cardinality proficiency. Together, these results suggest that children start to map nonsymbolic
representations onto newly acquired symbolic representations before they have mastered the
count sequence.
In contrast, other research suggests that number word mappings develop after children
understand the number system. Le Corre and Carey (2007) gave children aged 3-5 years a
count list elicitation task, a ‘Give-a-Number’ task, an estimation task and a nonsymbolic
ordinal judgment task. They classified children into number knower-levels based on the
highest number they reliably produced on the ‘Give-a-Number’ task. Children were either
classified as subset-knowers (one-knowers, two-knowers, three-knowers or four-knowers) or
cardinal principle-knowers (CP-knowers). Le Corre and Carey found that none of the subsetknowers and only half of the CP-knowers could map beyond four on the estimation task
without counting. They argue that the presence of a group of CP nonmappers suggests that
the mapping of number words beyond four is not part of the acquisition of the counting
principles. They further specify that number words beyond four are mapped onto
nonsymbolic numerosities about six months after children acquire the counting principles.
Data from Slusser and Sarnecka (2011) provides additional support for this view.
Children aged 2-4 years completed a ‘Give-a-Number’ task and a word extension task with
two alternative forced-choice options. The results revealed that only CP-knowers were above
chance at extending number words (“four”, “five”, “eight” and “ten”) from one set to another
based on numerosity. In contrast, all children succeeded at matching pictures by mood or
colour. In a follow-up experiment, Slusser and Sarnecka found that even CP-knowers had
difficulty extending words by number on trials where mood and colour were incongruent.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
11
These results further suggest that children do not map number words to nonsymbolic
representations until they understand the cardinal principle of counting.
Summarising these studies, there is inconsistent evidence regarding the relationship
between young children’s nonsymbolic skills and their knowledge of the counting system.
Specifically, we do not yet know whether young children can map number words to
nonsymbolic representations of quantity before, or only after they grasp the cardinality
principle of counting.
Methodological Issues
One reason for the inconsistent findings in the literature is that studies have used
different tasks to measure counting skills. These tasks tap into different aspects of children’s
counting knowledge – from number word sequence production to cardinality understanding –
and they vary in terms of their reliability. Specifically, it is not clear whether performance on
the ‘How many?’ task reflects any true conceptual understanding of cardinality. As noted
previously, it is thought to provide a weaker measure of cardinality than the ‘Give-a-Number’
task.
To assess the mapping between nonsymbolic-symbolic representations, researchers have
used symbolic comparison tasks, matching tasks or estimation tasks. The symbolic
comparison task is limited because it only provides an indirect measure of mapping skills.
Estimation tasks provide a more direct measure of mapping; however, they are also limited
due to the task demands placed on participants. Typically in these tasks participants are
presented with arrays of nonsymbolic (dot) stimuli and are asked to estimate the number of
dots. Studies have shown that, unless calibrated, adults’ estimates are generally inaccurate
(e.g., Izard & Dehaene, 2008) and many children fail to produce estimates at all (e.g., Lipton
& Spelke, 2005). The use of these free-response tasks with preschool children is therefore
questionable. Mundy and Gilmore (2009) developed a mapping task for use with school age
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
12
children which allows a more direct assessment of children’s mapping between symbolic and
nonsymbolic quantities. However, this still requires mapping skills to be assessed in both
directions (i.e. symbolic to nonsymbolic and nonsymbolic to symbolic) which increases the
number of trials, making it less suitable for use with young children. In view of these
difficulties, here we introduce a cross-notation comparison task in which preschool children
made magnitude comparison judgements across representational formats (e.g., dot arrays vs.
verbal number words). This allowed a direct assessment of the mapping between magnitude
representations that is suitable for use with young children.
Finally, a number of previous studies have been conducted with children who may have
had several months or years experience with counting. These studies cannot reveal the
processes involved in the very early stages of number acquisition. To determine whether
magnitude representations are involved in the process of learning to count, it is essential to
study these effects in children who are not yet experienced counters.
The Present Study
In this study we used a novel mapping task to investigate the link between preschool
children’s magnitude representations and their early counting skills. We gave children aged
2-4 years three tasks designed to measure separate competencies. Firstly, a count list
elicitation task was used to assess children’s mastery of the verbal count list. Secondly, a
‘Give-a-Number’ task was used to assess children’s cardinality understanding. Finally, a
cross-notation comparison task was used to assess children’s mapping between nonsymbolic
and symbolic magnitude representations. If, as found by Lipton and Spelke, (2005),
children’s ability to map number words to nonsymbolic numerosities is limited by knowledge
of the count list then we would expect performance on the count list elicitation task to be the
best predictor of performance on the cross-notation comparison task (Prediction 1).
Alternatively, performance on the cross-notation comparison task may be limited by
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
13
knowledge of cardinality. In this case we would expect only children who are CP-knowers on
the Give-a-Number task to be able to complete the cross-notation comparison task
(Prediction 2). Finally, if children map number words to quantities during the process of
acquisition and before they understand the cardinality principle then we would expect some
children who are not CP-knowers to be able to complete the cross-notation comparison task
(Prediction 3). If there is no evidence that children can complete the cross-notation
comparison task before they are CP- knowers then this suggests that symbolic-nonsymbolic
mappings do not play a critical role in acquiring the count sequence and thus these mappings
must be formed later, once children are proficient with the symbolic number system.
Alternatively, if children are able to map between symbolic and nonsymbolic representations
before they are CP knowers then this suggests that these mappings must form while children
are learning the number symbols, and may play a critical role in this process.
Method
Participants
Sixty preschool children (31 male) aged between 2 years 7 months and 4 years 10 months
(mean age 3 years 8 months) participated in the study. Children were recruited either through
local nurseries in Nottingham, UK, or through the University of Nottingham’s Human
Development and Learning participant database. Testing took place either at a child’s
nursery or at a university laboratory. Participation was voluntary and all children received
stickers for taking part. Children invited into the university also received a small gift to thank
them for taking part and caregivers received reimbursement for their travel expenses.
Seven children were excluded from all the analyses for the following reasons: English
was not their native language (n = 2), failure to attempt all tasks (n = 4), unable to produce an
accurate count list (n = 1). A further seven children attempted all tasks but completed
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
14
insufficient trials of the comparison task to allow analysis. These participants were excluded
from the analyses that involved the comparison task. Thus there were 46 complete datasets.
Tasks
Each child was presented with three tasks: a count list elicitation task (based on Le Corre
& Carey, 2007), a Give-a-Number task (based on Wynn 1990, 1992) and a novel crossnotation magnitude comparison task. These tasks allowed us to assess mastery of the verbal
count list, cardinality understanding and mapping between magnitude representations
(nonsymbolic and symbolic), respectively.
Count list elicitation task. The experimenter placed 20 small objects in a single row in
front of the child and asked them to count them aloud. The objects were identical in all
respects. Children were given a choice as to which character they wished to play the game
with and this character choice determined the set of objects used (felt strawberries, felt
bananas, plastic bricks).
If the child refused to produce a count list or if they produced a count list without
attending to the objects, the experimenter prompted them: “Can you count them with
me...one, two...,” whilst pointing at the objects. Each child was assigned a counting score
based on the highest number that they counted to correctly, regardless of whether they
pointed to the objects whilst doing so.
Give-a-Number task. Adapted from Wynn (1990, 1992) this task was presented in line
with previous studies (e.g., Condry & Spelke, 2008; Le Corre & Carey, 2007; Sarnecka &
Gelman, 2004). The experimenter placed 15 objects in a random cluster in front of the child.
The child was then asked to give their chosen character [character x] a given number [n] of
the objects. For example, “Can you give Mike the Monkey three bananas?” Following the
child’s response, the experimenter asked; “Is that [n]?” If the child responded with “yes”, the
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
15
experimenter proceeded with the next trial. If the child responded with “no”, the
experimenter repeated the original request.
The numbers 1-6 were requested in a pseudorandom order. The experimenter began by
asking the child to give 3 objects, and then increased or decreased the number accordingly.
Each number was requested up to three times; if a child succeeded once and failed once at a
given number, it was asked for a third time. Incorrect trials on which a child produced a
different number in the experimental range (1-6) were marked as incorrect for both the
requested and the produced number.
Children were categorised as knowers of a given number when they succeeded on two of
the three trials for that number (whilst also producing that number no more than half as often
when asked for a different number). Each child was assigned a number knower-level based
on the highest number that they reliably produced. For example, children who reliably
produced sets of 1 and 2 and 3 objects, but not 4 objects, were categorised as a threeknowers. Children who succeeded at the highest sets (5 and 6) were categorised as cardinal
principle-knowers (CP-knowers).
Cross-notation comparison task. In 12 experimental trials, children were instructed to
choose which of two characters had the most balls. The experimenter presented two
numerosities sequentially. The first numerosity was presented nonsymbolically as an array of
dots on a card (15.5cm x 10.5cm). As the experimenter presented the card they said:
“[Character x] has this many balls.” The second numerosity was presented symbolically as a
verbal number word, read aloud by the experimenter. To aid conceptualisation, the
experimenter presented a picture of a box on a card (the same size as the dot array cards) and
said: “[Character x] has hidden his/her balls in a box. [Character x] has [n] balls.” The
experimenter then asked: “Who has the most balls?” Both stimulus cards remained in place
next to each of the characters until the child gave their response (see Figure 1b). Children
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
16
responded by naming and/or pointing to the character. Children were instructed not to use
counting to solve this task and if the experimenter saw children begin to make any visible
counting acts, they were reminded not to.
The numerosities used ranged between 1-10. One third of the trials involved at least one
numerosity within the subitising range (1-3) and two thirds of the trials involved two
numerosities outside the subitising range (4-10). The numerical distance between the
quantities being compared was varied; half of the trials had a small numerical distance (a
distance of 1, 2 or 3) and half of the trials had a large numerical distance (a distance of 4 or
5). Each type of stimulus (dot array/verbal number word) was the larger amount an equal
number of times, and each character had the larger amount an equal number of times (see
Table 1). Dot arrays were generated randomly. All of the dots on a given stimulus card were
identical, however, colour (blue, green, red) and dot size (small, medium, large) were varied
across stimulus cards.
The experimenter presented Trials 1 and 2 successively, followed by Trials 3-12 in a
random order for each child. Trials 1 and 2 were presented first because they involved small
numbers (within the subitising range) only. Prior to the experimental trials, children
completed two nonsymbolic practice trials (dot array vs. dot array) to familiarise them with
the rules of the game. In Practice Trials 1 and 2 children were asked to compare 2 dots vs. 1
dot and 4 dots vs. 2 dots (see Figure 1a).
Procedure
Children were tested individually, in a single 10-20 minute session, either at their nursery
or a university laboratory. Children tested at nursery were accompanied by a member of the
nursery staff and children tested at the university were accompanied by their caregiver.
The experimenter sat opposite the child and presented all three tasks sequentially: 1)
Count list elicitation, 2) Give-a-Number and 3) Cross-notation comparison. The children
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
17
received a sticker after each task. The experimenter emphasised that they were number games
and children were encouraged to have a guess even if they were not sure. Throughout the
experiment, feedback on accuracy was not given; however, children received praise for their
general attention and behaviour.
Results
In presenting the results, we first provide a descriptive overview of the children’s
performance on each of the three tasks. In the sections that follow, we test the alternative
predictions derived from the previous literature. First we use correlations and regression to
explore the relationships among performance on each task and test whether count list
knowledge (Prediction 1) or cardinality knowledge (Predictions 2 and 3) was a stronger
predictor of performance on the cross-notation comparison task. In order to further tease out
the importance of full cardinal knowledge (Prediction 2) versus an immature cardinal
knowledge of a subset of numbers (Prediction 3) on performance on the cross-notation
comparison task, we consider performance on the cross-notation comparison task for groups
of children at different knower-levels in detail. Finally, we explore individual patterns of
performance across the cross-notation comparison and Give-a-Number tasks as a further test
of Predictions 2 and 3.
Children showed a range of performance on all of the tasks. The highest number reached
in the counting task ranged from 8 to 20 with a mean of 14.7 (SD = 3.9). Mean knower-level
on the Give-a-Number task was 3.85 (SD = 2.2, range 0 – 6). Mean accuracy on the crossnotation comparison task was 0.72 (SD = 0.2, range 0.3 – 1). This indicates that these tasks
were appropriate for use with this group of preschool children. Cronbach’s alpha for the
Give-a-Number task was 0.76 and for the cross-notation comparison task was 0.64. We
explored performance on the cross-notation comparison task to consider whether there was
any evidence that children were using covert counting to solve this task. Using a by-items
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
18
analysis we found that accuracy was not related to the size of the dot array (r = -.042, p =
.897) but was strongly related to the size of the verbal number word (r = -.860, p < .001).
These correlations were significantly different (t(9) = 3.74, p = .005). This suggests that
children were not using counting to solve this task, since young children’s counting
performance is related to set-size (Fuson, Pergament, Lyons, & Hall, 1985).
Table 2 reports the number of children achieving each knower-level on the Give-aNumber task. The distribution of knower-levels and ages is in line with previous research (Le
Corre & Carey, 2007). The absence of five-knowers in our sample suggests that once
children become four-knowers, they quickly move on to becoming cardinal principleknowers. The ages of our participants ranged from 2 years 7 months to 4 years 10 months, a
period that is rich in developmental change. Therefore we looked at whether children’s age in
months was related to their performance on each task. Age at testing was significantly related
to performance on all three tasks (Give-a-Number r = .80, p < .001; cross-notation
comparison r = .64, p < .001; count list elicitation r = .72, p < .001); thus we included age as
a covariate in our further analyses.
Correlations Among Tasks
We performed a series of correlations in order to explore relationships among the tasks.
There were positive correlations among children’s performance on each of the three tasks
(see Figure 2). Give-a-Number task knower-level was correlated with accuracy on the crossnotation comparison task, r = .71, p <.001 (controlling for age r = .41, p = .005) and with the
highest number reached in the count list elicitation task r = .70, p < .001 (controlling for age r
= .32, p = .022). Accuracy on the cross-notation task was also correlated with the highest
number reached in the count list elicitation task r = .58, p < .001 (controlling for age r = .268,
p = .079). These relationships were then subjected to partial correlations in order to explore
whether they represented independent relationships. Knower-level was still correlated with
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
19
accuracy on the cross-notation comparison task after controlling for count list knowledge r =
.53, p < .001 and knower-level was correlated with the highest number reached in the count
list elicitation task after controlling for accuracy on the cross-notation comparison task r =
.44, p = .003. However, accuracy on the cross-notation comparison task was no longer
correlated with the highest number reached on the count list elicitation task after controlling
for knower-level r = .22, p = .16. In keeping with this, in a linear regression predicting
performance on the cross-notation comparison task (r2 = .53, p < .001), knower-level was
found to be a significant independent predictor (β = .45, t = 2.50, p = .016) but count list
knowledge (β = .17, t = 1.09, p = .282) and age at testing (β = .18, t = 0.95, p = .346) were
not.
This analysis indicates that knower-level is a better predictor of cross-notation
comparison performance than count list knowledge or age, providing evidence against
Prediction 1 and in support of Prediction 2. Therefore, it is understanding of the cardinal
principle, rather than simply knowledge of the count sequence, which limits children’s ability
to map number words to nonsymbolic quantities.
Cross-notation Comparison Task Performance of Children at Different Knower-levels
If cardinal principle knowledge limits performance on the cross-notation comparison
task, is full cardinal principle understanding required for the task, in line with Prediction 2, or
is an immature cardinal understanding of only a subset of numbers sufficient, as suggested in
Prediction 3? We explored performance on the cross-notation comparison task by children at
different knower-levels, using a one-way ANOVA2 (see Figure 3). The participants were
grouped into one of three knower-level groups: one/two-knowers, three/four-knowers, and
CP-knowers. There was a significant effect of knower-level on cross-notation comparison
task accuracy, reflecting the fact that children who were at a higher knower-level were more
accurate, F(2, 42) = 25.0, p < .001. Bonferroni-corrected t-tests indicated that the mean
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
20
scores for the one/two-knowers (M = .54) and the three/four-knowers (M = .68) did not differ
significantly (p = .065), but the CP-knowers (M = .86) were significantly more accurate than
the one/two-knowers (p < .001) and the three/four-knowers (p = .008).
The cross-notation comparison task requires a choice to be made from two options for
each trial, and therefore chance performance was 50%. We explored whether children in each
of the groups performed above chance on the task using one-sample t-tests. CP-knowers and
three/four-knowers both performed above chance (t(22) = 11.93, p < .001 and t(7) = 3.41, p =
.011 respectively). However, one/two-knowers were not significantly above chance, t(13) =
1.28, p = .224.
These results suggest that children are able to map numerical symbols onto nonsymbolic
quantities once they have understood the cardinal meaning of some numbers (i.e. are at least
three/four-knowers), but the accuracy of the connection between numerical symbols and
nonsymbolic quantities improves once children are cardinal principle-knowers. This appears
to give a mixed picture regarding the relationship between these skills; children who are not
yet CP-knowers could perform the cross-notation comparison task to some extent, but their
performance was lower than the children who were CP-knowers, which does not provide
specific support for either Prediction 2 or 3.
However, in these analyses we considered all trials of the cross-notation comparison task.
For the children who were not yet CP-knowers this included some trials that were within their
knower-level, but also some trials which were within their count list but above their knowerlevel (e.g., asking a three-knower to compare six dots versus the number word “nine”). In
order to determine whether children who were not CP-knowers were able to connect number
words and nonsymbolic quantities within their knower-level range, we calculated a new set
of accuracy scores for those trials which contained only verbal numbers falling within the
knower-level of each participant. For example, for a participant who was a three-knower the
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
21
new accuracy score only included trials where the verbal number word ranged from one to
three. Therefore performance was considered across a different number of trials for children
at different knower-levels (12 trials for CP-knowers, 5 trials for four-knowers, 3 trials for
three-knowers and 2 trials for two-knowers). Note that there were no trials containing the
verbal number word “one” therefore one-knowers were excluded from the following
analyses.
We re-ran the ANOVA2 examining the effect of knower-level on performance on the
cross-notation comparison task. This time there was no effect of knower-level on accuracy,
F(2, 37) = .67, p = .519 (see Figure 3). We also used these new accuracy scores to compare
performance to chance level for both of the non-CP-knower groups. This time the three/fourknowers were found to be performing significantly above chance, t(7) = 3.12, p = .017, and
performance of the two-knowers approached significance t(8) = 2.29, p = .051.
These results suggest that, rather than only CP-knowers being able to complete the crossnotation comparison task in line with Prediction 2, children who are not yet CP-knowers are
accurate at mapping between nonsymbolic and symbolic representations when the verbal
number falls within their knower-level range. Thus there is evidence that children who do not
understand the cardinality principle are able to map number words that they understand the
cardinal meaning of to nonsymbolic representations, providing some support for Prediction 3.
If it is the case that both CP- and non-CP-knowers can map some verbal number words
onto nonsymbolic quantities, then, in line with theories of magnitude representations
(Feigenson et al., 2004), we would expect both groups to show a numerical distance effect in
their responses to the cross-notation comparison task. We therefore conducted a two-way
mixed design ANOVA with numerical distance (small, large) and knower-level (CP-knower,
non-CP-knower) as factors. Due to the small number of trials which fall within their knowerlevel, the one/two-knowers were not included in this analysis. As expected, there was a
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
22
significant effect of distance (F(1, 29) = 6.955, p = .013) with more accurate performance for
trials with a large (M = 0.85) compared to a small (M = 0.77) numerical distance, a signature
of the nonsymbolic system. Importantly, there was no significant interaction between group
and distance (F(1, 29) < 1, p = .510), indicating that both groups showed an equivalent
distance effect. The existence of the numerical distance effect in this task reflects the
approximate nature of the nonsymbolic system, providing evidence that the cross-notation
comparison task measures the ability to map between the two systems in both CP-knowers
and non-CP-knowers.
Individual Patterns of Performance
Finally, to further distinguish between Predictions 2 and 3, we explored individual
patterns of performance across the Give-a-Number and cross-notation comparison tasks. The
group-level analyses reported above suggest that, at a group-level, children were able to
complete the cross-notation comparison task within their knower-level even if they were not
CP-knowers on the Give-a-Number task. However, analysis of individual patterns of
performance can reveal how frequent this pattern of performance was. We therefore scored
the tasks on a pass/fail basis, scoring ‘1’ if they passed, and ‘0’ if they did not. Children were
considered to have passed the Give-a-Number task, if they were CP-knowers and were
considered to have passed the cross-notation comparison task if they scored above chance
level according to a binomial distribution (in the case of two-knowers if they answered all
items correctly). In line with the analysis above, we considered only the trials that fell within
an individual’s knower-level. We found that 9 children (21%) failed both the comparison task
and the Give-a-Number task and 16 children (38%) passed both tasks. Of the remaining
children, 8 (19%) passed the Give-a-Number task (i.e. were CP-knowers) but failed the
comparison task and 9 (21%) passed the comparison task but failed the Give-a-Number task
(i.e. were not CP-knowers). It is clear that there is no consistent order in which children were
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
23
able to pass the Give-a-Number and cross-notation comparison tasks, lending mixed support
for both Predictions 2 and 3.
Discussion
In this study we explored how children’s mappings between symbolic and non-symbolic
numerical representations are formed and whether they are involved in the acquisition of
symbolic skills. We found evidence that some children can map between nonsymbolic
representations and number words (within their knower level) before they are cardinal
principle knowers. This suggests that, for some children at least, these mappings form while
the symbolic system is being acquired, and may therefore play a role in this process.
Our results reveal that there is a complex relationship between count list knowledge,
understanding of the cardinality principle and the ability to map between number words and
nonsymbolic quantities. These findings have both theoretical and methodological
implications for the debate surrounding the role of nonsymbolic representations and
mappings between nonsymbolic and symbolic representations in the acquisition of symbolic
representations. Below we first review the theoretical conclusions that can be drawn from this
work before considering the methodological contributions. We end by considering the
implications for supporting the development of young children’s numeracy skills.
Theoretical Implications Regarding the Development of Symbolic-Nonsymbolic
Mappings
Contrary to Prediction 1, our results clearly demonstrate that it was knowledge of
cardinality, rather than count list knowledge, which was an important determinant of
children’s ability to map between symbolic and nonsymbolic representations. Cardinality
knowledge was related to mapping skill even after controlling for age. Previous studies have
shown that children may be able to accurately recite a count list some time before they grasp
the concept of cardinality (e.g., Wynn, 1990). We add to these findings by showing that count
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
24
list knowledge is also not sufficient for children to be able to map across symbolic and
nonsymbolic representations. Although children may learn the sequence of number words at
an early age, it is only when they have both grasped the concept of cardinality and mapped
these number words onto representations of magnitude that they can be considered to have
understood the meaning of their count list.
In contrast to Le Corre and Carey (2007) and Slusser and Sarnecka (2011) we found that
children could map between number words within their knower-level and nonsymbolic
quantities even if they did not grasp the general cardinality principle, in line with Prediction
3. Both group-level and individual-level analyses revealed that being a cardinal principleknower was not essential in order for children to correctly compare number words and
quantities that were within their knower-level. However, it does not appear that all children
necessarily develop these skills in a consistent order lending partial support to both
Predictions 2 and 3. Our findings in relation to Prediction 3 are tentative given that this was
based on the analysis of only a small number of trials. Future research should seek to
replicate this with studies in which the trials of the cross-notation comparison task are
carefully selected in light of children’s knower-level.
One possible explanation of the differences in our findings compared to the previous
literature might lie in the nature of the mapping task employed. The tasks used by Le Corre
and Carey (2007) required children to produce a verbal estimate for a nonsymbolic array. As
highlighted by Lipton and Spelke (2005), young children may be reluctant to make guesses in
a free estimate situation. Therefore, this task may have underestimated children’s knowledge.
Our cross-notation comparison task did not require free estimate production and thus the
general demands of the task were lower. In this way we were able to reveal children’s
mappings between number words and nonsymbolic quantities at an earlier age.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
25
The inconsistent patterns revealed by our individual-level analyses can be interpreted in
two alternative ways. First, it may be that cardinality understanding and the ability to map
between number words and quantities develop together but are initially fragile skills. As
children first acquire these skills there may be a period of time in which they are not able to
demonstrate this ability reliably in all contexts. A similar pattern is observed in the
development of children’s understanding of commutativity whereby task features affect
children’s early ability to demonstrate understanding of commutativity (Cowan, Dowker,
Christakis, & Bailey, 1996). Thus, for example, a child may show evidence of cardinality
understanding in one situation but not another. Over time this understanding becomes more
reliable across all contexts (c.f. Baroody, Wilkins, & Tiilikainen, 2003). Our pattern of
apparently inconsistent individual-level results may have arisen because some children were
unable to reliably reveal their understanding of cardinality or their ability to map between
number words and quantities.
On the other hand, it is possible that these findings reveal that there are alternative
developmental pathways in which children can acquire understanding of cardinality and the
ability to map number words to quantities. Some evidence suggests that there may be
individual differences in development pathways for other arithmetic concepts (e.g., Gilmore
& Papadatou-Pastou, 2009). For some children, knowledge of the symbolic number system
may take priority and they acquire understanding of the cardinality principle before they
make the connection between this system and the nonsymbolic system of magnitude
representations. For other children, the connection between the systems may take priority and
mappings develop at an earlier stage. Up until now, the existence of alternative pathways for
the development of early symbolic skills has yet to be considered (Le Corre & Carey, 2007;
Wagner & Johnson, 2011).
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
26
In order to distinguish between these possibilities, longitudinal research is needed in
which individual children’s performance is considered as they develop these two skills.
Given the rapid pace in which changes to early number understanding occur, detailed
microgenetic methods (Siegler & Crowley, 1991) would be appropriate to capture subtle
changes in children’s understanding. This would reveal whether all children follow the same
pathway but with fragility of skills as they appear, or alternatively whether groups of children
follow different developmental pathways.
Methodological Implications
In this study we introduced a new cross-notation comparison task in which children
compared symbols with nonsymbolic representations of quantity. This task allowed very
young children’s mapping between representations to be explored. This task does not suffer
from many of the limitations of existing dot comparison tasks (Inglis & Gilmore, 2014; Price,
Palmer, Battista, & Ansari, 2012). Since each trial only includes a single dot array, problems
with the visual characteristics of the arrays are reduced, and this reduces the inhibitory
control demands of the task (Fuhs & McNeil, 2013; Gilmore et al., 2013). Because the visual
characteristics of the arrays do not need to be controlled in the same manner as for dot
comparison tasks, a smaller number of trials is acceptable, making this task very suitable for
use with young children. We found no evidence that children used counting to solve this task
indicating that it is a good method to tap into the accuracy of children’s mapping between
symbolic and nonsymbolic representations.
We found that children’s performance on a count list elicitation task gave only a limited
picture of the importance of counting skills. Although there was a zero-order correlation
between count list knowledge and mapping skills, this was fully accounted for by the
relationship of both of these skills with cardinality understanding. Thus it was cardinality
understanding, rather than simply count list knowledge, that appeared to play an important
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
27
role in children’s broader mathematical development. This adds to the weight of evidence
that assessing count list knowledge provides only a limited picture of children’s
understanding of counting (e.g., Condry & Spelke, 2008; Wynn, 1995) and that cardinality
understanding is distinct from counting skill (Bermejo, Morales, & deOsuna, 2004). In
keeping with some previous research, our count elicitation task involved a set of objects for
children to count (e.g., Le Corre & Carey, 2007; Slaughter et al., 2006), while other studies
have asked children to count without a set of objects (e.g., Barth et al., 2009; Lipton &
Spelke, 2005). The differences in the demands of these task versions have not been fully
understood. It is thus important that researchers who are interested in the role of counting
skills in the development of broader mathematical achievement should employ a set of tasks
that capture the complex, multicomponential nature of counting skill.
Implications for Supporting the Development of Early Number Skills
Our findings show that there is a complex interplay between two distinct skills in young
children’s numeracy development. The first of these involves making connections between
number words and quantities. Giving meaning to abstract numerical symbols is crucial for
children’s later success with mathematics (Mundy & Gilmore, 2009) as it is the mapping
between symbols and magnitudes that gives numbers their meaning. For some children at
least, making the connection between numbers and magnitudes may actually help them come
to understand the symbolic system itself, i.e. to understand cardinality. This suggests that
activities which help children to link number words and quantities may be beneficial in
promoting their understanding of the symbolic system. Activities such as guessing games, in
which children estimate approximate numerosities, or board games (e.g., Siegler & Ramani,
2008), in which they match symbols and magnitudes, may be ways to help children connect
symbols and quantities.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
28
While some children appear to develop mapping skills first, our findings suggest that for
other children, gaining sophisticated understanding of the symbolic counting system takes
priority. Therefore, alongside activities to promote the connection between numbers and
quantities, children should be given the opportunity to develop their understanding of the
symbolic counting system itself. Simple activities that involve challenging children to reflect
upon the number of items in a set have been found to successfully enhance children’s
understanding of cardinality (Bermejo et al., 2004). Engaging with a wide variety of activities
involving counting may thus help children along the path to a rich understanding of numbers
and all their meanings.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
29
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Footnotes
1. Note that in the case of small numbers (less than 4), success may depend on set size
rather than ratio. There is ongoing debate as to whether small numbers engage the same
approximate number system as larger numbers (see Feigenson, Dehaene & Spelke, 2004, for
an overview of the two core systems hypothesis).
2. These results are replicated when using non-parametric statistics.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
36
Table 1
Cross-Notation Comparison Problems
Quantity 1 Quantity 2 Problem Number
Distance
(dot array) (verbal number word) 1 Small 1 3 2 Small 3 2 3 Large 1 5 4 Large 6 2 5 Small 5 4 6 Small 7 5 7 Large 4 8 8 Large 5 10 9 Small 6 9 10 Small 7 10 11 Large 9 4 12 Large 10 6 Note. Quantity 1 indicates the number of dots presented on the card and Quantity 2 indicates
the verbal number word given.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
37
Table 2
Number of Children at Each Knower-Level According to Performance on the Give-a-Number
Task and Mean Age of Children in Each Group
Knower level
Number of children Mean age
0
2
36.5
1
8
38.1
2
10
38.0
3
4
40.0
4
5
48.2
Cardinal Principle
23
51.0
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
Figure 1
38
Experimental Set-up of the Cross-Notation Comparison Task.
a)
“Upsy Daisy has this many
“Iggle Piggle has this
balls”
many balls”
“Upsy Daisy has this many
“Iggle Piggle has hidden
balls”
his balls in a box. Iggle
“Who has the most balls?”
b)
“Who has the most balls?”
Piggle has ten balls”
Figure1: Part a) gives an example of a nonsymbolic (dot comparison) practice trial and
part b) gives an example of a cross-notation comparison trial.
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
Figure 2
Scatterplots Depicting the Relationships Among Performance on the Three
Tasks
a
20
r = .70, p < .001
Highest count list
18
16
14
12
10
8
0
1
2
3
4
Give-a-number knower level
5
6
b
Cross-notation comparison accuracy
1.0
0.8
r = .71, p < .001
0.6
0.4
0.2
1
2
3
4
Give-a-number knower level
5
6
c
Cross-notation comparison accuracy
1.0
0.8
0.6
r = .58, p < .001
0.4
0.2
8
9
10
11
12
13
14
15
16
Highest count list
17
18
19
20
21
Figure 2: a) Give-a-Number task and count list elicitation b) Give-a-Number task and
cross-notation comparison task and c) cross-notation comparison task and count list
elicitation.
39
Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS
1.0
1.0
Accuracy for all trials
Accuracy
40
Accuracy for trials within knower-level
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
1,2 knower
3,4 knower
Knower Level
CP knower
0
1,2 knower
3,4 knower
Knower Level
CP knower
Figure 3. Performance on the cross-notation comparison task according to knower-level
showing accuracy on the full set of trials and the subset of trials which fall within knowerlevel range.